Transcript One-Way Analysis of Variance

Repeated Measures
ANOVA
Starting with One-Way RM
More fascinating than a
bowl of porridge. Really
KNR 445
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ANOVA II
Slide 2
One-Way Repeated Measures ANOVA
 Data considerations
 One continuous dependent variable (Likert
type data also acceptable)
 One nominal independent variable of > 2
levels
 Analogous to dependent t-test, but for more
than 2 levels of the independent variable
KNR 445
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ANOVA II
Slide 3
One-Way Repeated Measures ANOVA
 Advantages of repeated measures
 Again, as per paired t-test...
 Sensitivity
 Reduction in error variance (subjects serve as
own controls)
 So, more sensitive to experimental effects
 Economy
 Need less participants
 With many levels, this might be even more
important for ANOVA than t-test
 (need to be careful of fatigue effects, though)
KNR 445
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ANOVA II
Slide 4
One-Way Repeated Measures ANOVA
 Possible uses of 1-way RM ANOVA
 Same people measured 3+ times
 Pre-test, post-test, follow-up
 Same people measured under three or
more different treatments
 Drug 1, drug 2, drug, 3
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ANOVA II
Slide 5
One-Way Repeated Measures ANOVA
 Possible serious disadvantage of RM
 Order effects and treatment carry-over
effects (goes for paired t-test too)
 Should counterbalance (by random
assignment to order of treatment)
 E.g. (for 2 levels of RM: A & B)
Order of
administration
% of
subjects
1
2
50
A
B
50
B
A
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ANOVA II
Slide 6
One-Way Repeated Measures ANOVA
 Possible serious disadvantage of RM
 Order effects and treatment carry-over
effects (goes for paired t-test too)
 E.g. (for 3 levels of RM: A, B & C)
This type of control
for order effects is
known as a Latin
Square design
Order of administration
% of
subjects
1
2
3
33
A
B
C
33
B
C
A
33
C
A
B
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ANOVA II
Slide 7
One-Way Repeated Measures ANOVA
 Possible serious disadvantage of RM
 Treatment carry-over effects (goes for
paired t-test too)
 Even if order effects are controlled for,
there must be sufficient time between
treatments so that you can be sure that
the score on each level of the RM is due to
only one treatment (not a combination of
two or more)
 Note – order & treatment carry-over
effects are design rather than statistical
issues, but very important nevertheless
KNR 445
FACTORIAL
ANOVA II
Slide 8
One-Way Repeated Measures ANOVA
 Example, with chat about variance
partitioning and assumptions...
 Remember the one-way between
subjects ANOVA?
 Data looked like this in SPSS
 And the trick was to make variance
due to treatments bigger than
variance due to everything else (&
everything else included variance due
to individual differences)
 Well, what if you could take out
variance due to individual
differences?
KNR 445
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ANOVA II
Slide 9
One-Way Repeated Measures ANOVA
 That’s what the one-way RM
ANOVA does
 Data now looks like this, as
each person is measured on all
levels of the IV
 Variation due to individual
differences can then be
separated from variation due
to chance, as the same people
are present within each
condition.
 It goes something like
this...(cue horror movie music)
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ANOVA II
Slide 10
One-Way Repeated Measures ANOVA
Here’s the output from the between
subject ANOVA.
Note the size of the error term
(within groups variance measure).
That really diminishes the F-size.
And no-one likes a small F-size.
KNR 445
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ANOVA II
Slide 11
One-Way Repeated Measures ANOVA
 Now a pause before we
consider variance
partitioning in RM ANOVA, as
we see how to conduct the
wee devil. Suffice to say I’ll
be keeping it simple.
 Here’s the first step
Choose
this...
And you’ll
get this
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ANOVA II
Slide 12
One-Way Repeated Measures ANOVA
 Now...
You have to specify the independent
variable, and the number of levels it
has (4 here)
Then click “add” and
proceed by clicking
“define”
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Slide 13
One-Way Repeated Measures ANOVA
 Then...
...next (long process here) you
choose all the levels of the repeated
measures factor (i.v.)...
And slide them over to the
“within subjects variables” box
– just another name for
repeated measures
variable...or factor
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ANOVA II
Slide 14
One-Way Repeated Measures ANOVA
 And...output!
This first bit is from the multivariate
(more than one dependent variable – 4
here) approach to repeated measures. It
has some potential advantages (essentially
that one does not have to meet the
sphericity assumption...see next slide)
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ANOVA II
Slide 15
One-Way Repeated Measures ANOVA
 And...more output...
This bit is important. It’s a test of one of the more important assumptions of
RM ANOVA – sphericity. It’s kind of like the homogeneity of variance test, but
it’s the variance of the difference scores between the levels of the
independent variable that are being tested…you really have to adjust for it,
& we see how on the next slide (if this is NOT significant, it’s good)
Another important
bit…the HuynhFeldt Epsilon...see
next slide
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Slide 16
One-Way Repeated Measures ANOVA
 And...still more output...
Finally, the bit that counts. Note there are FOUR (count ‘em) separate
versions of each effect. Here’s the rule (Schutz & Gessaroli, 1993): If HuynhFeldt Epsilon (see previous slide) is > .7, use Huynh-Feldt adjusted F (third
line). If it is less than .7, use G-G (second line)
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ANOVA II
Slide 17
One-Way Repeated Measures ANOVA
 Same bit once again...
Here, you can see that, as the epsilon is 1, there is no correction, and the F
statistic stays the same throughout.
Now, what about that variance partitioning???? Remember we were going to
talk about that?
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ANOVA II
Slide 18
One-Way Repeated Measures ANOVA
 One last bit (that you can ignore)...
Let’s just look at this first. In the top
box, you can see a bunch of stuff like
“linear”, “quadratic”, & “cubic” –
that’s to do with the shape of the
difference that the change in scores
might take as they progress from drug
1 to drug 4, and only really makes
sense in trend analysis, which is again
beyond our scope.
Finally, down here you see “between
subjects effects”. There are none here
(just one I.V., and it’s RM). The error
variance here is essentially a measure
of individual differences, as we’ll see
in a minute...
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ANOVA II
Slide 19
One-Way Repeated Measures ANOVA
 So, how does the variance thing work?
 Let’s compare the two methods (“between
subjects” and “repeated measures”)
directly, bearing in mind where the
variances in the output tables come from
 In this way, my goal is simply to indicate
the benefit of taking out variation due to
individual differences
 We’ll start with the between subjects
method...(see next slide)
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Slide 20
One-Way Repeated Measures ANOVA
Here the between groups variance is 698.2
– this is just variation of mean scores on
the different treatments about the overall
mean...so this is the bit that is essentially
the treatment effect
26.4
25.6
15.6
32
And here is the within subjects variation...it
is calculated from the sum of the variation
within each of the treatments about each of
the treatment means...so it’s like a
combination of variation due to individual
differences, variation due to treatments, and
variation within treatment across individuals
(what?)
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Slide 21
One-Way Repeated Measures ANOVA
 Now for the repeated measures version:
27
16
23
34
24.5
Sum of squares = 680.8
(= sum of squared deviations from
the mean of these 5 scores, which
is 24.9, multiplied by the # levels
of the I. V.))
Note that the average score for each
subject across the four treatments is
different. This is due to individual
differences...and is the “between
subjects” error variance
 The thing that makes
repeated measures
powerful is that this
variation is taken out of
the within subjects error
term...see next slide
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ANOVA II
Slide 22
One-Way Repeated Measures ANOVA
An example from
excel
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Slide 23
One-Way Repeated Measures ANOVA
Now what you have to
see is that the SS for the
denominator in the F test
in RM is now 112.8, which
is derived from
793.6 – 680.8 = 112.8
individual differences
Error variation in
between subjects
ANOVA
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Slide 24
One-Way Repeated Measures ANOVA
And finally, as a direct
consequence of all this, the
numerator in the F-test is
unchanged (698.2), but the
denominator has been
reduced from 49.6 to 9.4,
resulting in an increase in F
from 4.69 to 24.76!
which of course
means...more
significance, more
power
KNR 445
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ANOVA II
Slide 25
One-Way Repeated Measures ANOVA
 So, to summarize
 Because of the way RM ANOVA partitions
variance for the RM factors, we have a far
more powerful test for the RM factors
 But you have to make sure you control for
spurious effects by controlling for order
effects and carryover effects
KNR 445
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ANOVA II
Slide 26
Example of interpretation of results
Note partial η2 is reported too
 Interpretation:
 A one-way repeated measures ANOVA was conducted to
student’s confidence in statistics prior to the class,
immediately following the class, and three months after the
class. Due to a mild violation of the sphericity assumption (
= .82), the Huynh-Feldt adjusted F was used. There was a
significant difference in confidence levels across time, F
(1.421, 41.205) = 33.186, p < .001, partial η2 = .86.
Dependent t-tests were used as post-hoc tests for significant
differences with Bonferroni-adjusted  = .017. Confidence
levels after three months (M = 25.03, SD = 5.20) were
significantly higher than immediately following the class (M =
21.87, SD = 5.59), which in turn were significantly higher
than pre-test levels (M = 19.00, SD = 5.37).
KNR 445
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ANOVA II
Slide 27
ANOVA/Inferential Statistics Wrap-up
 Inferential tests to compare differences in
groups:










Independent t-tests 
Dependent t-tests 
One-way ANOVA

Factorial ANOVA

One-way repeated measures ANOVA

Factorial repeated measures ANOVA
Mixed between-within groups ANOVA (split-plot)
Analysis of covariance (ANCOVA)
Multivariate analysis of variance (MANOVA)
Nonparametric tests (next)
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Slide 28
Factorial RM ANOVA
 Same notions as for factorial ANOVA –
main effects, interactions and so on
 Data set up a bit tricky
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ANOVA II
Slide 29
Two-way ANOVA with repeated
measures on one factor
 Sometimes referred to as a split plot or
Lindquist type 1 or (most commonly in my
experience) a “Two-way ANOVA with
repeated measures on one factor.”
 Research question: Which diet (traditional,
low carb, exercise only) is more effective in
weight loss across three time periods (before
diet, three months later, six months later)?
Is there a weight loss across time?
KNR 445
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ANOVA II
Slide 30
Two-way ANOVA with repeated
measures on one factor
 Diet RQ (continued)
 Looks like a 3x3 two-factor ANOVA, except
that one of the factors is a repeated
measure (one group of subjects tested
three times)
 As such, a two-factor between-groups
ANOVA is not appropriate; rather, we have
one factor that is between diet types and
another that is within a single group of
subjects
KNR 445
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ANOVA II
Slide 31
Two-way ANOVA with repeated
measures on one factor
 Use a mixed design ANOVA when:
 A nominal between-subjects IV with 2+
levels
 A nominal within-subjects IV with 2+
levels
 A continuous interval/ratio DV
 Note: you can add additional IV’s to this
test, but just as with Factorial ANOVA,
when you get 3+ IV’s, interpreting findings
gets really nasty due to all of the
interactions
KNR 445
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ANOVA II
Slide 32
Two-way ANOVA with repeated
measures on one factor
 Interactions: like a two-factor
between-subjects ANOVA, there may
be both main effects for each of the
two IV’s plus an interaction between
the two IV’s
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Slide 33
Analysis of Covariance
 An extension of ANOVA that allows you to explore
differences between groups while statistically
controlling for an additional continuous variable
 Can be used with a nonequivalent groups pretest/post-test design to control for differences in
pre-test scores with pre-existing groups
 You could use a mixed design ANOVA here, but with small
sample size, ANCOVA may be a better alternative due to
increased statistical power
 Be careful of regression towards the mean as a cause of
post-test differences (after using the covariate to adjust
pre-test scores)
KNR 445
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ANOVA II
Slide 34
Multivariate Analysis of Variance
 An extension of ANOVA for use with multiple
dependent variables
 With multiple DV’s, you could simply use
multiple ANOVA’s (one per DV), but risk
inflated Type 1 error
 Same reason we didn’t conduct multiple t-tests
instead of an ANOVA
 Ex. Do differences exist in GRE scores and
grad school GPA based on race?
 There are such things as Factorial MANOVA’s,
RM MANOVA’s, and even MANCOVA’s
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Slide 35
Finito!